论文题目：粗几何上的指标问题的局部化方法
作者简介：王勤，男,1967年08月出生，1997年02月师从于复旦大学陈晓漫教授，于2000年07月获博士学位。
摘
要
粗几何上的指标理论是“非交换几何”领域九十年代以来发展起来的重要研究方向，它孕育于非紧流形上的指标理论，其主要目标是通过几何空间（如非紧完备黎曼流形、有限生成群等）的大尺度几何结构探索指标代数，即Roe代数，的K-理论群的信息，从而建立几何空间的几何、拓扑与分析之间的联系，并应用于解决其他重要问题，如Novikov猜测、Gromov-Lawson-Rosenberg正标量曲率猜测、群C*-代数幂等元问题等。用粗几何的观念研究非紧空间上的指标问题，这种想法来源于指标定理的热方程方法的启发。事实上，非紧流形上的广义椭圆算子的K-理论指标并不依赖于流形的局部几何，而是依赖于流形的大尺度几何结构，即流形的粗几何。在几何空间上通过控制局部紧算子的传播速度产生的C*-代数，即Roe代数C*(X)，恰好反映了几何空间的粗结构特征；广义椭圆算子的指标就是落在Roe代数的K-群之中。从几何空间的一个容易计算的几何不变量，即粗化K-同调群
，到Roe代数的K-理论群
有一个指标映射，粗Baum-Connes猜测断言这个指标映射为同构，从而提供了计算Roe
代数K-理论群的有效途径。因此，粗几何上的指标理论的中心问题就是解决粗Baum-Connes猜测。
1993年，J.
Roe 引入粗上同调理论与Roe代数，研究了粗几何同指标理论的联系。1995年，G.
Yu（郁国梁），N. Higson和J. Roe分别用不同的方法给出了粗Baum-Connes猜测的表述形式，并对一些具体的空间证明了该猜测。1997年，G.
Yu受热方程方法的启发，引入局部化Roe代数
，把粗Baum-Connes猜测的证明归结为证明从粗化K-同调群到局部化Roe代数的K-理论群的局部指标映射为同构。本篇博士论文的研究内容正是受局部化Roe代数的启发而确立的。一方面，我们同时受K-同调的Paschke对偶理论的启发，引入了关于拟局部算子的局部化C*-代数
，并结合局部化Roe代数
，建立了K-同调的局部化对偶理论及其同粗Baum-Connes猜测的联系。另一方面，我们意识到上述局部化方法可以看成是随着时间趋向无穷而对几何算子的传播速度进行局部化的方法，这进一步启发我们通过在空间的无穷远处对几何算子的传播速度进行局部化的方法来研究指标问题。用这种空间局部化方法，我们构造了新的指标代数
并研究了相应的指标定理。此外，在计算Roe代数的K-理论群的过程中，Roe代数的理想往往扮演着重要的角色。因此，本文还刻画了Roe代数的理想结构。
全文共分四章，第一章介绍了粗几何上的指标理论的基本概念和相关结论。
第二章从对偶的角度发展局部化的思想，建立了局部化K-同调理论及其同粗Baum-Connes猜测的联系。我们首先引入关于拟局部算子的局部化C*-代数
；
是
的一个双边闭理想。受N. Higson和J. Roe于1995年所刻画的Paschke对偶理论的启发，定义几何空间X的局部化K-同调群为
。我们证明了局部化代数具有稳定性；局部化代数的K-理论群与局部化K-同调群在粗几何范畴上具有函子性质和强Lipschitz同伦不变性。在K-理论的计算中，切割与粘贴技术通常是行之有效的手段。对于几何空间的强切割分解，我们建立了局部化代数的K-理论群与局部化K-同调群的Mayer-Vietoris循环正合序列。利用这些性质和计算工具，我们研究了局部化K-同调群同Kasparov解析K-同调群的关系，证明了在有限维非紧单纯复形上，两种K-同调群同构。从这个结论出发我们进而推出了把Roe代数、局部化Roe代数、它们的对偶代数的K-理论群以及粗化K-同调群、局部化K-同调群联系在一起的一个正合双复形，并由此双复形证明了：有界几何空间上的粗Baum-Connes猜测成立的充分必要条件是局部化代数
中的一个理想的K-群
为零。这部分结果发表在J.
London Math. Soc., 66(2002)227—239上。
局部化代数包含着比较丰富的可用于计算的信息，从理论上看，
的K-群相对易于计算。为了说明这一点，我们对离散度量空间上的局部化代数的K-理论群进行了详细的计算，从而证明了相应的局部化指标映射同构。这部分结果即将发表于The
Southeast Asian Bulletin of Mathematics。
第三章利用在空间无穷远处的局部化方法研究了关于粗几何的指标问题。我们首先引入一个新的指标C*-代数
，它由那些在空间无穷远处传播速度趋向零的局部紧算子生成，是几何空间的渐近粗化过程的非交换商。结合该代数的对偶代数
，我们证明了几何空间的K-同调群同构于商代数
的K-理论群，从而用在空间无穷远处的局部化代数重新刻画了K-同调的对偶理论。由此产生了从K-同调群
到指标代数
的K-理论群的渐近指标映射。我们证明了这些K-同调群和K-理论群具有渐近Lipschitz同伦不变性；对于渐近可标度的几何空间（包括欧氏锥、单连通非正曲率完备黎曼流形等），证明了渐近指标映射为同构。这部分结果已发表在Manuscripta
Mathematica, 110, 475—485
(2003)。由该杂志审稿人的意见，我们了解到上面建立的指标问题完全可以用一般的粗结构的框架来表述；上述渐近指标映射的同构问题，正是几何空间关于渐近粗结构的粗Baum-Connes猜测。作者曾应邀在2002国际数学家大会的“算子代数及其应用”卫星会议上报告过本文的结果。有趣的是在大会期间，我们被G.
Yu、J. Roe告知N. Wright（当时为Roe的博士生）用和我们同样的想法对具有有限渐近维数的几何空间的粗Baum-Connes猜测给出了新的证明（这个结果最早是1998年由G.
Yu证明的），他的这个结果即将发表在J. Functional Analysis上。
第四章研究了关于Roe代数的理想结构的若干问题，此前国内外文献中还没有这方面的结果发表。Roe代数具有相当复杂的代数结构；但有一类理想支撑于几何空间X的子空间Y，通常记为C*(Y,
X)，在Roe代数K-理论群的计算中非常有用。G. Yu于1997年在复旦讲学期间问：是否Roe代数的所有理想都具有C*(Y,
X)的形式？在本章中我们证明了Roe代数的可数生成理想都不具有C*(Y,
X)的形式，并给出一个理想具有C*(Y, X)形式的充分必要条件，以及建立了Roe代数理想同几何空间的子空间的粗等价类的保序对应关系，刻画了Roe代数的理想都不具有可数逼近单位元等性质。这些结果的证明，展示了Roe代数结构的复杂性不仅来自几何空间在无穷远处的拓扑行为，而且来自Roe代数所作用的Hilbert空间的局部无穷维性以及二者的联合作用。这部分结果已发表于Oxford
Quarterly J. Math., 52(2001)437—446。
在1999—2000年间，粗几何上的指标理论的研究取得了重大进展，一方面，G
Yu利用局部化技术结合其他工具对一致可嵌空间证明了粗Baum-Connes猜测，从而对相当广泛的空间类证明了Novikov猜测，并引出C*-代数理论的正合问题与Novikov猜测的有趣联系。另一方面，Gromov、Higson、Yu用膨胀图上Roe代数中的鬼投影元给出了粗Baum-Connes猜测的反例。鬼投影的出现，使人们意识到对粗 Baum-Connes猜测的研究将任重而道远。我们从理想结构的角度研究了鬼投影现象，证明了Roe代数中的算子为鬼元的充分必要条件是该算子的主理想中的有限传播算子都是紧算子；具有G.
Yu引入的(A)性质的几何空间的Roe代数中没有鬼元。我们还引入“近似膨胀图”的几何构造，展示了鬼元产生的广泛途径。作者曾应邀在2002国际数学家大会的15分钟报告中介绍了这些结果。论文的摘要发表在ICM2002
Abstrasts of Short Communications and Poster Sessions, 9 (161—162)
2002。在进一步的研究中，我们用一类超滤子完全刻画了单核空间上一致Roe代数的极大理想结构，论文已投于Chinese
Annals of Mathematics。进而引入控制切割技术，证明了一致Roe代数的包含稠密的有限传播算子的理想格、粗结构的理想格、粗空间的理想格、粗几何广群的单位空间的不变开子集格互相同构，从而对理想的几何构造作出了比较系统、清晰的刻画，并利用Schur乘子的技术证明了具有(A)性质的几何空间的Roe代数的所有理想中有限传播算子都稠密等结论，论文投于J.
Functional Analysis。
Abstract
The
index theory associated with coarse geometry is an important research field of
“Noncommutative Geometry”, developed in the 1990’s from the index theory
on noncompact manifolds. Its main goal is to explore information in the K-theory
groups of the index C*-algebras, the
Roe algebras C*(X), by using the large-scale geometrical structure of proper metric
spaces, including noncompact complete Riemannian manifolds, finitely generated
groups, etc., so as to establish connections among geometry, topology and
analysis of the geometric spaces, and furthermore, to solve other relating
problems, say, the Novikov conjecture, the Gromov-Lawson-Rosenberg conjecture on
positive scalar curvature, the idempotent problem in the theory of C*-algebras.
The idea of coarse geometry is motivated by the heat equation approach to the
index theorem on noncompact manifolds. In fact, the K-theory indices of
geometric operators does not depend on the local geometry of noncompact
manifolds, but on the large scale geometry, i.e., the coarse geometry, of the
spaces. The Coarse Baum-Connes Conjecture proposes a formula for computation of
the K-theory groups of the Roe algebras, the container of indices of geometric
operators, by the coarse K-homology groups of the geometric spaces, which are
computable invariants.
In
1993, J. Roe introduced a coarse cohomology theory and a C*-algebra,
called the Roe algebra later, to study index theorems on noncompact manifolds in
spirit of coarse geometry. In 1995, G. Yu, N. Higson and J. Roe succeeded in
formulating the Coarse Baum-Connes Conjecture in different ways, respectively,
and verified it for several examples. Motivated by the heat equation methods, G.
Yu introduced in 1997 the localization Roe algebras
, and shown that, to attack the Coarse Baum-Connes Conjecture, it suffices to
prove that the local index map is an isomorphism from the coarse K-homology to
the K-theory of the localization Roe algebras. The research in the present Ph.
D. dissertation is mainly inspired by these localization Roe algebras. Firstly,
combining the idea of the Paschke duality for K-homology and the idea of
localization, we introduce the localization algebra of pseudolocal operators and
establish the duality for the localization version of K-homology and its
connection with the Coarse Baum-Connes Conjecture. Secondly, we understand that
the above approach is a method of localization as time tends to infinity. This
inspires us to use the method of localization at infinity of space to study the
relevant index problems of coarse geometry. Consequently, we produce a new index
C*-algebra C*A
(X) and develop the associated index theorems. Finally, we also
characterize the ideal structure of the Roe algebras since it plays an important
role in the calculation of K-theory.
The
dissertation consists of four chapters. In the first chapter, we present basic
notions and relevant results in the index theory associated with coarse
geometry.
In
the second chapter, we introduce the localization algebra
for pseudolocal operators and, according to the Paschke duality for K-homology
characterized by N. Higson and J. Roe in 1995, define the localization
K-homology as
for geometric spaces X.
We show that these localization algebras are stable
C*-algebras; the K-theory of these localization algebras and the K-homology
of the spaces are functorial in the coarse category. They are also invariants
under the strong Lipschitz homotopy. Usually, the trick of cutting and pasting
is very effective in computation of K-theory. We show that, for the strong
excision of proper metric spaces, there are cyclic exact sequences for the
K-theory of the localization algebras and the K-homology of the spaces. By using
these tools, we are able to investigate the relation between the localization
K-homology and the classical Kasparov analytic K-homology. It turns out that
these two K-homologies are the same things over the finite dimensional
simplicial complexes with spherical metric. It follows that the Coarse Baum-Connes
Conjecture can be fixed by showing that the K-theory of an ideal in the
localization algebra
is trivial. These results have been published in J.
London Math. Soc., 66(2002)227—239. We also give a detailed
calculation for the K-theory groups of these localization algebras over
uniformly discrete proper metric spaces and show that the corresponding
localization index maps are isomorphisms. These results have been accepted to
publish in The
Southeast Asian Bulletin of Mathematics soon.
In
the third chapter, we use the method of localization at infinity of space to
study index theorems with coarse geometry. There is a new index C*-algebra
, generated by locally compact operators whose propagation tends to zero at
infinity of space. It is also regarded as the noncommutative quotient of
asymptotic coarsening of the geometric spaces. We show that the K-homology
groups of the spaces are isomorphic to the K-theory groups of the quotient
, where
is the dual algebra of
. That is, the Paschke duality can also be retrieved by these new localization
algebras. This characterization results in the asymptotic index maps from the
K-homology
to the K-theory of
. We establish an asymptotic Lipschitz homotopy invariance theorem for these
K-homology groups and K-theory groups. We show that the asymptotic index maps
are isomorphisms for the asymptotically scaleable spaces, which include
Euclidean cones, simply connected complete Riemannian manifolds with nonpositive
curvature. These results have been published in Manuscripta
Mathematica, 110, 475—485 (2003)。According
to the referee’s report, we realized that the index theory we formulated above
fits into the general framework of coarse category. The index theorem for the
asymptotic index maps is nothing but the coarse Baum-Connes conjecture for the
asymptotic coarse structure of the geometric spaces. The author has bees invited
to report this work at the ICM 2002 Satellite Conference on Operator Algebras
and Applications. It is interesting to mention that the author was told by G. Yu
and J. Roe during ICM that N. Wright, who is then a Ph. D. student of J. Roe,
had just developed the same ideal as ours and give a new proof for the coarse
Baum-Connes conjecture over spaces with finite asymptotic dimension (this result
was originally proved by G. Yu in 1998). His work will appear in J. Functional
Analysis.
In
the fourth chapter, we study the ideal structure of the Roe algebras. There is a
class of ideals supported on subspaces of the geometric space, which are very
useful in computation of K-theory. During his visit at the Institute of
Mathematics, Fudan University, in 1997, G. Yu asked whether or not all ideals in
the Roe algebras are supported on subspaces? We give a negative answer to this
question in this chapter by showing that none of the countably generated ideals
are supported on subspaces. We also find a necessary and sufficient condition
for an ideal to be supported on a subspace, and establish an order preserving
bijection between the collection of ideals supported on subspaces and the
collection of coarse equivalence classes of subspaces. Other related properties
are also discussed. The proofs of these results reveal that the complicated
structure of the Roe algebras results from not only the topological behaviors at
infinity of the metric spaces but also the local infinity of the Hilbert spaces
on which the Roe algebras act. These results have been published in Oxford
Quarterly J. Math., 52(2001)437—446.
During
1999—2000 several great achievements were accomplished on index theory with
coarse geometry. On one hand, G. Yu proved the coarse Baum-Connes conjecture,
and hence the Novikov conjecture, for
uniformly embeddable spaces, a quite broad scope of spaces. This result also
leads to the discovery of the interesting connection between the exactness
conjecture in the theory of C*-algebras
and the Novikov conjecture. On the other hand, M. Gromov, N. Higson, G. Yu found
counterexamples to the coarse Baum-Connes conjecture over expander graphs, the
Roe algebra of which contains a ghost projection. This ghost projection predicts
that there will be a quite long way to go to completely solve the coarse Baum-Connes
conjecture. We probe the ghost phenomenon by ideal structure, showing that an
operator in the Roe algebra is a ghost if and only if the finite propagation
operators in the principle ideal of the operator are compact. If the space has
G. Yu’s property (A), the Roe algebra contains no ghosts. We also construct an
almost expander to show that there are many ways for ghosts to arise. The author
has introduced these results at a short invited talk at ICM 2002 held in
Beijing. The abstract of the paper was published in ICM2002
Abstrasts of Short Communications and Poster Sessions, section 9
(161—162) 2002. In another work of this aspect, we completely characterize the
structure of maximal ideals in the uniform Roe algebra over simple corses by
using a kind of ultrafilters. This paper is submitted to Chinese Annals of Mathematics.
Recently, we find a technique of controlled truncation on operators and show
that the following four lattices are mutually isomorphic: the ideals in which
finite propagation operators are dense, the invariant open subsets of the unit
space of the groupoid of coarse structure, the ideals of the coarse structure
and the ideals in the coarse spaces. Thus, the geometrical constructions for
ideals of the uniform Roe algebras have been characterized clearly and
systematically. This work has just been submitted to J. Functional Analysis.
